Find all values of $ x $ where$ f\left(x\right) > 0 $.XXXYYY-4-4-4

The function is positive for all values of $ x $.

Analyze the Graph: Finding Where f(x) > 0

Q: Why isn't x = -4 included if the graph touches the x-axis there?

Great observation! At $ x = -4 $, the function equals zero, not a positive value. We need $ f(x) > 0 $, which means strictly greater than zero.

Q: How do I know the function is positive on both sides of x = -4?

Look at the graph carefully! The curve is above the x-axis for all x-values except exactly at $ x = -4 $ where it just touches the axis. Above the x-axis means positive values.

Q: What's the difference between f(x) > 0 and f(x) ≥ 0?

$ f(x) > 0 $ means strictly positive (excludes zero), while $ f(x) ≥ 0 $ means non-negative (includes zero). For this problem, we exclude $ x = -4 $ because we want strictly positive.

Q: Can I write the answer as x ≠ -4 instead?

Not quite! While $ x ≠ -4 $ excludes the right point, it doesn't specify that we want all other real numbers. The correct notation is $ x or \( x > -4 $.

Q: How can I verify this answer makes sense?

Pick test points! Try $ x = -5 $ (left of -4) and $ x = 0 $ (right of -4). If the graph shows positive y-values at these points, your answer is correct!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Find all values of $x$

where $f\left(x\right) > 0$ .

2

Step-by-step solution

In this problem, we are tasked with determining the values of $x$ for which the function $f(x)$ is positive. We have been provided a graphical representation of the function, and we will use this graph to find our solution.

1. Restate the problem: We need to find all values of

x

where the function

f(x)

is greater than zero, based on its graphical representation. 2. Identify key information: The graph is typically that of some function

f(x)

. The graph shows points and lines that illustrate where the function is above and below the x-axis. Points or curves on or above the x-axis indicate positive values. 3. Potential approach: Analyze where the graph is above the x-axis. 5. The most appropriate approach is to visually inspect the graph to identify when the curve is above the x-axis. 6. Steps needed: - Identify any turning points or intersections with the x-axis. - Determine the segments of the x-axis where the function is above it. 8. Simplify the inspection by focusing on intervals separated by intersections with the x-axis. 9. Consider that the function might only touch the x-axis at specific points, like at roots, and analyze behavior around these points.

Based on the graph, we observe the following behavior of the function $f(x)$ :

The function intersects the x-axis at $x = -4$ . This indicates a potential root or turning point where the function transitions from positive to negative or vice versa.
From the graph, it appears that the function is above the x-axis on both sides of $x = -4$ , except exactly at $x = -4$ , where it touches the x-axis.

Hence, the function $f(x)$ is positive for $x > -4$ and for $x < -4$ . Note that exactly at $x = -4$ , the function is zero, not positive.

Therefore, the solution is: $x > -4$ or $x < -4$ .

In conclusion, the function $f(x)$ is positive for these values of $x$ , except the point where it touches the x-axis.

The corresponding choice given the problem's options is:

$x > -4$ or $x < -4$

3

Final Answer

$x > -4$ or $x < -4$

Key Points to Remember

Essential concepts to master this topic

Visual Rule: Function is positive when graph lies above x-axis
Technique: Check intervals separated by x-intercept at $x = -4$
Check: Test points: left of -4 gives positive, right of -4 gives positive ✓

Common Mistakes

Avoid these frequent errors

Including the x-intercept in the solution
Don't include x = -4 in your answer when finding where f(x) > 0! At x = -4, the function equals zero, not greater than zero. Always exclude points where the graph touches or crosses the x-axis when finding positive regions.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below intersects the X-axis at points A and B.

The vertex of the parabola is marked at point C.

Find all values of $$ x $$ where $f\left(x\right) > 0$ .

FAQ

Everything you need to know about this question

Why isn't x = -4 included if the graph touches the x-axis there?

+

Great observation! At $x = -4$ , the function equals zero, not a positive value. We need $f(x) > 0$ , which means strictly greater than zero.

How do I know the function is positive on both sides of x = -4?

+

Look at the graph carefully! The curve is above the x-axis for all x-values except exactly at $x = -4$ where it just touches the axis. Above the x-axis means positive values.

What's the difference between f(x) > 0 and f(x) ≥ 0?

+

$f(x) > 0$ means strictly positive (excludes zero), while $f(x) ≥ 0$ means non-negative (includes zero). For this problem, we exclude $x = -4$ because we want strictly positive.

Can I write the answer as x ≠ -4 instead?

+

Not quite! While $x ≠ -4$ excludes the right point, it doesn't specify that we want all other real numbers. The correct notation is $x < -4$ or $x > -4$ .

How can I verify this answer makes sense?

+

Pick test points! Try $x = -5$ (left of -4) and $x = 0$ (right of -4). If the graph shows positive y-values at these points, your answer is correct!

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Analyze the Graph: Finding Where f(x) > 0

Graph Analysis with Sign Determination

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Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why isn't x = -4 included if the graph touches the x-axis there?

How do I know the function is positive on both sides of x = -4?

What's the difference between f(x) > 0 and f(x) ≥ 0?

Can I write the answer as x ≠ -4 instead?

How can I verify this answer makes sense?

🌟 Unlock Your Math Potential

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